The effect of partial incoherence on the modulational instability (MI) of two optical waves that interact through cross-phase modulation caused by the Kerr nonlinearity is analyzed by using the Wigner approach. The problem is solved as an initial value problem providing a complete description of the time evolution of the wave perturbations. It is shown that the partial incoherence tends to suppress the MI of the two interacting waves by giving rise to a damping effect directly determined by the characteristic coherence length of the stochastic process.

The propagation of partially incoherent light in nonlinear media is analyzed using the Wigner transform method. The power and versatility of this approach is illustrated by several examples which clearly demonstrate how partial incoherence tends to suppress coherent instabilities by weakening the nonlinearity. In particular, it is found that the effect of partial incoherence on modulational instabilities can be described in terms of a Landau-like damping effect, which counteracts the coherent growth rate of the instability. Similarly, in the case of the self-focusing collapse instability, the nonlinear focusing effect becomes successively smaller as the coherence length of the light decreases and eventually no collapse phenomenon occurs.

A statistical approach based on the Wigner transform is proposed for the description of partially incoherent optical wave dynamics in nonlinear media. An evolution equation for the Wigner transform is derived from a nonlinear Schrödinger equation with arbitrary nonlinearity. It is shown that random phase fluctuations of an incoherent plane wave lead to a Landau-like damping effect, which can stabilize the modulational instability. In the limit of the geometrical optics approximation, incoherent, localized, and stationary wave fields are shown to exist for a wide class of nonlinear media.

A statistical multistream description of quantum plasmas is formulated, using the Wigner-Poisson system as dynamical equations. A linear stability analysis of this system is carried out, and it is shown that a Landau-like damping of plane wave perturbations occurs due to the broadening of the background Wigner function that arises as a consequence of statistical variations of the wave function phase. The Landau-like damping is shown to suppress instabilities of the one- and two-stream type.

An investigation is made of the effects of initial or dynamically induced chirping on the linear propagation characteristics of pulses in optical fibers. It is shown that under certain conditions the chirping will give rise to a splitting of the initial pulse into two separating subpulses. The necessary conditions for pulse splitting to occur are established analytically and corroborated by numerical computations.

A novel statistical approach based on the Wigner transform method is proposed for the description of partially incoherent optical wave dynamics in nonlinear media. The Wigner-Moyal equation is derived for the Wigner distribution of the optical wave field governed by the nonlinear Schrodinger equation with an arbitrary nonlinearity. An application to incoherent light propagation in dispersive Kerr media shows that random phase fluctuations of a plane wave solution lead to a linear Landau-like damping effect, which can stabilize the nonlinear modulational instability,. A similar effect is shown to occur in the case of the two-stream instability of two partially incoherent optical waves interacting with cacti other through cross-phase modulation initiated by the nonlinearity In the limit of the geometrical optics approximation, it is shown that ID and 2D self-trapped. stationary and incoherent wave pulse structures may exist for a wide class of nonlinear media. Furthermore, time-dependent self-similar ID and 2D structures have been found in the case of the Kerr nonlinearity.

An investigation is made of ultrafast pump–probe pulse collisions near the zero-dispersion wavelength in an optical single-mode fiber. A steplike probe frequency shift is observed when the pump power is gradually increased. The magnitude of this frequency jump is shown to depend on the phase difference between the pulses. This new effect is investigated numerically and experimentally and is attributed to four-wave mixing.

Incoherent light propagation in dispersive nonlinear Kerr media is studied using a statistical description based on the Wigner transform. It is demonstrated that the incoherence leads to a linear Landau-like damping effect, which can stabilize the modulational instability.

An analytical as well as numerical analysis is made of the dynamics induced on a weak signal wave pulse by a co-propagating strong pump wave pulse in a nonlinear Kerr medium. Emphasis is given to the situation where the group velocity dispersion and the non-linearity of the signal pulse have opposite signs. In this defocusing situation, it is demonstrated that the pump splits the signal pulse into two frequency shifted pulse fragments which separate in time, asymptotically with constant separation velocity. Explicit analytical predictions for the velocity of separation are obtained and corroborated by numerical simulations.

We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein condensate (BEC) trapped in an external potential. As an example, we consider a parabolic double-well potential and derive coupled-mode equations for the complex amplitudes of the BEC macroscopic collective modes. Our equations describe different regimes of the condensate dynamics, including the nonlinear Josephson effect for any separation between the wells. We demonstrate macroscopic self-trapping for both repulsive and attractive interactions, and confirm our results by numerical simulations.